Problem: Factor the following expression: $ x^2 - xy - 42y^2 $
Solution: When we factor a polynomial of this form, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + ay)(x + by)&=&xx &+& xby + ayx &+& ayby \\ \\ &=& x^2 &+& {(a+b)}xy &+& {ab}y^2 \\ &\hphantom{=}& \hphantom{x^2} &\hphantom{+}& \hphantom{{-1}xy} &\hphantom{+}& \hphantom{{-42}y^2} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + ay)(x + by)}&\hphantom{=}&\hphantom{xx} &\hphantom{+}& \hphantom{xby + ayx} &\hphantom{+}&\hphantom{ayby} \\ &\hphantom{=}& \hphantom{x^2} &\hphantom{+}&\hphantom{{(a+b)}xy}&\hphantom{+}&\hphantom{{ab}y^2} \\ &=& x^2 &+& {-1}xy &+& {-42}y^2 \end{eqnarray} $ The coefficient on the $xy$ term is $-1$ and the coefficient on the $y^2$ term is $-42$ , so to reverse the steps above, we need to find two numbers that add up to $-1$ and multiply to $-42$ You can start by trying to guess which factors of $-42$ add up to $-1$ . In other words, you need to find the values for $a$ and $b$ that meet the following conditions: $ {a} + {b} = {-1}$ $ {a} \times {b} = {-42}$ If you're stuck, try listing out every single factor of $-42$ and its opposite as $a$ in these equations, and see if it gives a value for $b$ that validates both conditions. For example, since $6$ is a factor of $-42$ , try substituting $6$ for $a$ as well as $-6$ The two numbers $6$ and $-7$ satisfy both conditions: $ {6} + {-7} = {-1} $ $ {6} \times {-7} = {-42} $ So we can factor the polynomial as $(x + 6y)(x - 7y)$.